The original Constellation wasn't a model as such. (Definitely not the kind of model that was used for the Enterprise.) AFAIK, it was basically a cheap plastic toy. Something like that could be quite easily kitbashed, especially since they never ended up using it again.
Erm, no, it was an AMT Enterprise model, hurridedly damaged and redressed to be the Constellation in the week's time they had to get the VFX for the episode filmed. It was a cheap model, yes, but it was not a toy by any stretch of the imagination.
Still, though, they could have kitbashed the hell out of it and come out ahead, since I can only guess they never planned on using the thing again.
The remastering could also have fixed the problem. Either change the ship itself to make it obviously a different class, or alter its registry number.